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Extensions of Szegö's theory of orthogonal polynomials, II

Research paper by Attila Máté, Paul Nevai, Vilmos Totik

Indexed on: 01 Dec '87Published on: 01 Dec '87Published in: Constructive Approximation



Abstract

Let {ϕn(dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofϕn(dμ) when logμ'∈L1. In what follows we will discuss the asymptotic behavior of the ratio φn(dμ1)/φn(dμ2) off the unit circle in casedμ1 anddμ2 are close in a sense (e.g.,dμ2=g dμ1 whereg≥0 is such thatQ(eit)g(t) andQ(eit)/g(t) are bounded for a suitable polynomialQ) and μ1′>0 almost everywhere or (a somewhat weaker requirement) limn→∞Φn(dμ1,0)=0, for the monic polynomials Φn. The consequences for orthogonal polynomials on the real line are also discussed.